Optimal. Leaf size=139 \[ \frac {2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{3} a^2 c^3 x^{3/2}+\frac {2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac {2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac {2}{23} b^2 d^3 x^{23/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {448} \begin {gather*} \frac {2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{3} a^2 c^3 x^{3/2}+\frac {2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac {2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac {2}{23} b^2 d^3 x^{23/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 448
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 \sqrt {x}+a c^2 (2 b c+3 a d) x^{5/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{9/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{13/2}+b d^2 (3 b c+2 a d) x^{17/2}+b^2 d^3 x^{21/2}\right ) \, dx\\ &=\frac {2}{3} a^2 c^3 x^{3/2}+\frac {2}{7} a c^2 (2 b c+3 a d) x^{7/2}+\frac {2}{11} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{11/2}+\frac {2}{15} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{15/2}+\frac {2}{19} b d^2 (3 b c+2 a d) x^{19/2}+\frac {2}{23} b^2 d^3 x^{23/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 139, normalized size = 1.00 \begin {gather*} \frac {2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{3} a^2 c^3 x^{3/2}+\frac {2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac {2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac {2}{23} b^2 d^3 x^{23/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.07, size = 163, normalized size = 1.17 \begin {gather*} \frac {2 \left (168245 a^2 c^3 x^{3/2}+216315 a^2 c^2 d x^{7/2}+137655 a^2 c d^2 x^{11/2}+33649 a^2 d^3 x^{15/2}+144210 a b c^3 x^{7/2}+275310 a b c^2 d x^{11/2}+201894 a b c d^2 x^{15/2}+53130 a b d^3 x^{19/2}+45885 b^2 c^3 x^{11/2}+100947 b^2 c^2 d x^{15/2}+79695 b^2 c d^2 x^{19/2}+21945 b^2 d^3 x^{23/2}\right )}{504735} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.85, size = 130, normalized size = 0.94 \begin {gather*} \frac {2}{504735} \, {\left (21945 \, b^{2} d^{3} x^{11} + 26565 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + 33649 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + 168245 \, a^{2} c^{3} x + 45885 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + 72105 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.38, size = 135, normalized size = 0.97 \begin {gather*} \frac {2}{23} \, b^{2} d^{3} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c d^{2} x^{\frac {19}{2}} + \frac {4}{19} \, a b d^{3} x^{\frac {19}{2}} + \frac {2}{5} \, b^{2} c^{2} d x^{\frac {15}{2}} + \frac {4}{5} \, a b c d^{2} x^{\frac {15}{2}} + \frac {2}{15} \, a^{2} d^{3} x^{\frac {15}{2}} + \frac {2}{11} \, b^{2} c^{3} x^{\frac {11}{2}} + \frac {12}{11} \, a b c^{2} d x^{\frac {11}{2}} + \frac {6}{11} \, a^{2} c d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, a b c^{3} x^{\frac {7}{2}} + \frac {6}{7} \, a^{2} c^{2} d x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} c^{3} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 138, normalized size = 0.99 \begin {gather*} \frac {2 \left (21945 b^{2} d^{3} x^{10}+53130 a b \,d^{3} x^{8}+79695 b^{2} c \,d^{2} x^{8}+33649 a^{2} d^{3} x^{6}+201894 a b c \,d^{2} x^{6}+100947 b^{2} c^{2} d \,x^{6}+137655 a^{2} c \,d^{2} x^{4}+275310 a b \,c^{2} d \,x^{4}+45885 b^{2} c^{3} x^{4}+216315 a^{2} c^{2} d \,x^{2}+144210 a b \,c^{3} x^{2}+168245 a^{2} c^{3}\right ) x^{\frac {3}{2}}}{504735} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.03, size = 127, normalized size = 0.91 \begin {gather*} \frac {2}{23} \, b^{2} d^{3} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {19}{2}} + \frac {2}{15} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {15}{2}} + \frac {2}{3} \, a^{2} c^{3} x^{\frac {3}{2}} + \frac {2}{11} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {11}{2}} + \frac {2}{7} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 119, normalized size = 0.86 \begin {gather*} x^{11/2}\,\left (\frac {6\,a^2\,c\,d^2}{11}+\frac {12\,a\,b\,c^2\,d}{11}+\frac {2\,b^2\,c^3}{11}\right )+x^{15/2}\,\left (\frac {2\,a^2\,d^3}{15}+\frac {4\,a\,b\,c\,d^2}{5}+\frac {2\,b^2\,c^2\,d}{5}\right )+\frac {2\,a^2\,c^3\,x^{3/2}}{3}+\frac {2\,b^2\,d^3\,x^{23/2}}{23}+\frac {2\,a\,c^2\,x^{7/2}\,\left (3\,a\,d+2\,b\,c\right )}{7}+\frac {2\,b\,d^2\,x^{19/2}\,\left (2\,a\,d+3\,b\,c\right )}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.36, size = 155, normalized size = 1.12 \begin {gather*} \frac {2 a^{2} c^{3} x^{\frac {3}{2}}}{3} + \frac {2 b^{2} d^{3} x^{\frac {23}{2}}}{23} + \frac {2 x^{\frac {19}{2}} \left (2 a b d^{3} + 3 b^{2} c d^{2}\right )}{19} + \frac {2 x^{\frac {15}{2}} \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{15} + \frac {2 x^{\frac {11}{2}} \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right )}{11} + \frac {2 x^{\frac {7}{2}} \left (3 a^{2} c^{2} d + 2 a b c^{3}\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________